The Madoff case, coming on the heels of a tentative $13 billion settlement over JPMorgan’s mortgage practices, poses another major threat to the reputation of the nation’s largest bank.

Reflecting the magnitude of the investigation, prosecutors and JPMorgan have held preliminary discussions about a so-called deferred prosecution agreement, people briefed on the inquiry said. Such an arrangement would suspend criminal charges against JPMorgan in exchange for a fine, certain other concessions and an acknowledgment that the bank will face charges if it fails to behave. Prosecutors may also require JPMorgan, which has repeatedly said that “all personnel acted in good faith” in the Madoff matter, to hire an independent monitor.

While deferred-prosecution agreements are the Justice Department’s preferred tool for punishing corporate giants — they allow prosecutors to appear tough without imperiling a company’s health — they are typically deployed only when misconduct is severe. For a large American bank, they are nearly unheard-of.

But the government, the people added, has not ruled out a harsher punishment for JPMorgan Chase’s national banking subsidiary. Prosecutors could demand that the unit plead guilty to a criminal violation of the Bank Secrecy Act, a federal law requiring financial institutions to report suspicious activity to the government.

Underscoring concerns that a guilty plea could destabilize the bank, the people said, prosecutors have discussed the ramifications of criminal charges with one of JPMorgan’s regulators. But the regulator, the Office of the Comptroller of the Currency, assured the prosecutors that it would not interfere.

Representatives for JPMorgan, the Comptroller and the prosecutors declined to comment. Authorities could announce an action by the end of the year, the people briefed on the inquiry said, a move that could cap a multiyear investigation. Prosecutors, the people said, are weighing criminal charges against JPMorgan employees who did business with Mr. Madoff. It is unclear which employees are under investigation.

The investigation, led by the F.B.I. and the United States attorney’s office in Manhattan, centers on whether JPMorgan failed to alert federal authorities to Mr. Madoff’s conduct. JPMorgan served as Mr. Madoff’s primary bank for more than two decades, giving it a unique window onto his practices.

The case will most likely hinge on a series of e-mails that suggest JPMorgan continued to work with Mr. Madoff even as questions mounted about his operation. In one e-mail that surfaced in a separate lawsuit, a JPMorgan employee acknowledged that Mr. Madoff’s outsize returns seemed “a little too good to be true.”

The people briefed on the inquiry, who spoke on the condition of anonymity because they were not authorized to discuss private negotiations, cautioned that the government had not decided to charge any current or former JPMorgan employees. Likewise, the discussions with the bank itself are preliminary and the government has not concluded what action to take. Two of the people noted that prosecutors were more likely to seek a deferred prosecution agreement than to demand a guilty plea.

Neither JPMorgan nor any other big Wall Street bank has ever been subjected to such an agreement before, according to a University of Virginia Law School database. Among large American banks, only Wachovia and the banking arm of American Express have entered into such an agreement.

But if it does pursue a guilty plea, the government would deal another blow to the reputation of JPMorgan and its chief executive, Jamie Dimon. The bank was once an industry favorite in regulatory circles.

The actual repercussions would depend on the underlying criminal charge. The most serious potential violation could complicate JPMorgan’s business with certain clients, possibly forcing investors like pension funds to withdraw some money from the bank. But a lesser violation would be likely to have more of a reputational consequence.

For the government, it would represent an extraordinarily rare show of force. Ever since a criminal indictment led to the demise of the accounting firm Arthur Andersen, Enron’s auditor, the government has been wary of imposing criminal charges on big corporations for fear that it would imperil the institution and have ripple effects on the broader economy. Under federal guidelines, prosecutors must weigh “collateral consequences,” like job losses and economic implications, in such an action.

HSBC, for example, paid $1.9 billion to settle a money-laundering case, but the Justice Department stopped short of indicting the British bank. The case reinforced concerns that big banks, having grown so large and interconnected, are too big to indict.

Yet Preet Bharara, the United States attorney in Manhattan whose office is handling the JPMorgan case, has disputed that theory. In a recent speech, Mr. Bharara said he rejected the idea from companies that “because we’re so big, to take action against us, the sky is going to fall.”

“I don’t think anyone is too big to indict — no one is too big to jail,” Mr. Bharara said at another speech.

The Manhattan United States attorney and the F.B.I are not the only ones pursuing JPMorgan over the Madoff case. The Office of the Comptroller of the Currency recently sent the bank a notice indicating that the agency would soon fine the bank over the Madoff case, two people briefed on the case said.

The developments come at a difficult time for JPMorgan, which faces an onslaught of government scrutiny.

The tentative $13 billion settlement in the mortgage case would resolve an array of state and federal investigations into the bank’s sale of trouble mortgage investments. The bank, authorities suspect, sold mortgage securities in the run-up to the financial crisis without fully warning investors of the risks.

JPMorgan is also grappling with an investigation into the bank’s decision to hire the sons and daughters of senior Chinese government officials. And Mr. Bharara’s office is examining whether some of the bank’s trading in the energy markets amounted to manipulation.

The Madoff case is particularly thorny. Any action would link the bank to the most notorious financial criminal in more than a generation. Mr. Madoff orchestrated a Ponzi scheme lasting decades that wiped out an estimated $17 billion in cash for his investors. Paper losses reached more than $64 billion.

Mr. Madoff is serving a 150-year sentence in a federal prison in North Carolina after pleading guilty in March 2009. In a 2011 interview from prison, Mr. Madoff told The New York Times that the banks he did business with “had to know.”

Mr. Madoff’s ties to JPMorgan trace to 1986, when it became his primary banker. Over the course of that relationship, Mr. Picard claims, JPMorgan “made at least half a billion dollars in fees and profits” from the relationship.

The bank, according to Mr. Picard’s lawsuit, generated handsome sums by allowing Mr. Madoff’s brokerage firm to “funnel billions of dollars” through its account with JPMorgan, “disregarding its own anti-money laundering duties.”

The bank, starting around 2006, also pursued derivatives deals linked to Mr. Madoff’s so-called feeder-fund investors, the hedge funds that invested their clients’ money with him.

About that time, concerns began to circulate within JPMorgan.

“I do have a few concerns and questions,” one JPMorgan employee wrote in February 2006 after studying some of Mr. Madoff’s trading records, according to an e-mail cited in the lawsuit. “All trades are generated by Madoff’s black box.”

But JPMorgan’s derivatives deals, which allowed investors to collect returns tied to the profits of the feeder funds, took off anyway. By June 2007, JPMorgan’s “Equity Exotics” unit had sold more than $130 million worth of the deals to investors, Mr. Picard’s lawsuit said.

That month, JPMorgan employees sought approval to push the total to $1.32 billion, according to the lawsuit.

On June 15, 2007, when a JPMorgan committee met to ponder the proposal, new suspicions emerged about Mr. Madoff. A senior risk management officer at the bank e-mailed colleagues to report that another bank executive “just told me that there is a well-known cloud over the head of Madoff and that his returns are speculated to be part of a Ponzi scheme.” The senior officer added that “I think we owe it to ourselves to investigate further.”

But according to Mr. Picard, the bank’s further research amounted to a phone call with Mr. Madoff and “a Google search with no follow-up.”

Similar concerns were enough to deter JPMorgan’s own private bank from doing business with Mr. Madoff. In an e-mail, a JPMorgan wealth management executive remarked that Mr. Madoff’s “Oz-like signals” were “too difficult to ignore.”

After Mr. Madoff’s arrest in December 2008, Mr. Picard said, a flurry of JPMorgan e-mails captured the lack of surprise at the bank.

One employee, referring to the agenda for the June 2007 meeting, wrote, “Perhaps best this never sees the light of day again!!”